Dynamic unbalance calculating method and dynamic unbalance testing device

ABSTRACT

The magnitudes and angular positions of dynamic unbalance in rotor are determined by the calculating method of this invention from the original vibration vectors. The original vibration vectors (magnitude and angular position) obtained by measurements using a vibration sensor at bearings supporting the rotor has its specific unbalance calculated from the dynamic unbalance vectors (magnitude, angular position) in the rotor obtained by the vector calculation method on the basis of the ratios of the distances between the bearings to the distance between the two correction planes chosen arbitrarily in the rotor.

FIELD OF APPLICATION

This invention relates to a calculating method to determine the dynamicunbalance vectors in rotors.

PRIOR ART

Up to this time, the calculating method to determine the dynamicunbalance in rotor of rotating machines is as follows:

The original vibration vectors (displacement, velocity, or acceleration)are measured at bearings effected by the centrifugal force due tounknown dynamic unbalance in the rotor. Then the machine is stopped anda trial mass is attached on the rotor. Subsequently the machine is ranand the vibration vectors changed by the trial mass are measured. Thesetest runs with trial mass are necessary to repeat at several times.Using the measured quantities, the dynamic unbalance in the rotor can bedetermined by the calculating with conversion coefficient method andsuch.

ABSTRACT OF THE DISCLOSURE

The object of this invention is to eliminate the burdens which theordinal method needs several times stops to run, disassemblings,attachings with trial mass, reassemblings, retest runs and measurings todetect the changed vibration vectors, and to produce the calculatingmethod to determine the dynamic unbalance in rotor from only theoriginal vibration vectors measured under the initial operatingcondition without a trial mass.

The feature of this invention is to calculate the dynamic unbalancevectors (magnitudes and directions) in rotor, by thegeometric-vector-calculation using the ratios for distance lengthsbetween bearings and the distance length between the correction planes,on the view point of this invention that the vector quantities(displacement, velocity, or acceleration) measured at both bearingsunder the initial operating condition, corresponding to the centrifugalforces due to the dynamic unbalance of rotor represented at the portionsof journals.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a image drawing for the example of carrying out theinvention.

FIG. 2 shows a example of carrying out the invention by the vectorcalculation in case of a₁=a₂.

FIG. 3 shows a image drawing for example of carrying out the inventionby the vector calculation in case of a₁≠a₂.

FIG. 4 shows a example of carrying out the invention by thevector-computationin case of a₁≠a₂.

DESCRIPTION OF THE MARKS

-   -   a₁, a₂: Length between the correction planes and bearings.    -   b: Length between the correction planes.    -   l₁, l₂: Length between the center of gravity and bearings.    -   B₁,B₂: Bearings supporting the journals of rotating shaft.    -   L: Length between the bearings    -   R: Main portion of rotor    -   G: Gravity-center of rotor    -   I, II: Correction planes (dynamic balance planes)    -   X: X-coordinate, basis of angle for rotor    -   Y: Y-coordinate    -   Z: Z-coordinate, center of rotating axis (main shaft)

BEST MODE OF CARRYING OUT THE INVENTION

Followings are the descriptions of a best mode of carrying out theinvention taken in connection with the accompanying drawings.

1. FIG. 1 (a three-dimensional drawing) shows a model for the state ofthat the most general rotor, R, of the rotating machines supported inlevel by the bearings B1,B2, to be applied the invention

The unbalance centrifugal forces caused by the dynamic unbalance inrotor, transfer to the bearings B1,B2 through the journals of the mainshaft (z-coordinate) and the vibrations can be produced on the bearingsB1,B2.

For the vibrations on the bearings, the vibration vector quantities(magnitude & direction for displacement, velocity, or acceleration)measured in X-coordinate are X₁,X₂.

In case of that the vibration vectors are the amplitudes ofdisplacements X₁, X₂ [cm], the equation of motion for the forcedvibrations of the centrifugal forces, F₁, F₂ acting to the bearingsB1,B2 which due to the unbalance in rotor, can be expressed by thefollowing differential equation: $\begin{matrix}{{{M_{1,2}\frac{\mathbb{d}^{2}x_{1,2}}{\mathbb{d}t^{2}}} + {c_{1,2}\frac{\mathbb{d}x_{1,2}}{\mathbb{d}t}} + {k_{1,2}x_{1,2}}} = {F_{1,2}\cos\quad\omega\quad{t\quad\lbrack N\rbrack}}} & {1\text{-}1}\end{matrix}$where,

-   -   M_(1,2): inertia mass acting to the bearings B1,B2, [Ns²/cm]        provided that, $\begin{matrix}        {{M_{1} = {M\frac{I_{2}}{L}}},{M_{2} = {M{\frac{I_{1}}{L}\quad\left\lbrack {{Ns}^{2}/{cm}} \right\rbrack}}}} & {1\text{-}2}        \end{matrix}$    -   M: mass of rotor [Ns²/cm]    -   C_(1,2): viscous damping coefficient of the bearings B1,B2        [N·s/cm]    -   k_(1,2): stiffness of the bearings B1,B2 in x-coordinate [N/cm]        k _(1,2)=(ω n ₁ ,n ₂)²M_(1,2) [N/cm]  1-3    -   ω n₁,n₂: natural frequencies of the bearings B1,B2 in        x-coordinate [rad/s]    -   ω: angular velocity of rotor [rad/s]    -   t: time [s]

The magnitudes of the unbalance centrifugal forces F₁, F₂ in rotor canbe obtained by the following equation based on the solution of the Eq.1-1F _(i) =k _(i) x _(i)√{square root over ([1−(ω/ω_(ni))²]²+[2(c _(i) /c_(ci))(ω/ω_(ni))]²)}[N]  1-4where,

-   -   i: 1, 2, to be continued followings    -   C_(ci): critical damping coefficient of the bearings B1,B2 in        x-coordinate [N·s/cm]

2. As the alternate solution for the magnitudes of the unbalancecentrifugal forces F₁, F₂, the unbalances distributing in the rotor canbe represented at the intensive and arbitrary two planes (defined ascorrection planes or dynamic balance planes) being vertical to therotationally axis (main shaft), and on the view-point of this inventionthat “the unbalance centrifugal forces in rotor acting to the bothbearings, are equivalent to the unbalance centrifugal forces caused inthe dynamic balance planes of the both portions of the journals”,because of the journals of rotor supported by the bearings are also tobelong to the portion of rotor, so the magnitude [N] of the unbalancecentrifugal forces F₁, F₂ can be expressed by the following equationF _(1,2) =M _(1,2) e _(1,2)ω² [N]  2-1where, e_(1,2): specific unbalances at the bearing B1,B2 (journals)planes [cm]

3. The amount of specific unbalance, e_(1,2), can be obtained by thefollowing equation of motion which the Eq. 2-1 is substituted in Eq. 1-1$\begin{matrix}{{{M_{1,2}\frac{\mathbb{d}^{2}x_{1,2}}{\mathbb{d}t^{2}}} + {c_{1,2}\frac{\mathbb{d}x_{1,2}}{\mathbb{d}t}} + {k_{1,2}x_{1,2}}} = {M_{1,2}{e\quad}_{1,2}\omega^{2}\cos\quad\omega\quad{t\quad\lbrack N\rbrack}}} & {3\text{-}1}\end{matrix}$Based on the solution of Eq. 3-1, the amount of specific unbalance,e_(1,2), can be obtained by the following equation $\begin{matrix}{e_{1} = {x_{1}{\frac{\sqrt{\left\lbrack {1 - \left( {\omega/\omega_{ni}} \right)^{2}} \right\rbrack^{2} + \left\lbrack {2\left( {c_{i}/c_{oi}} \right)\left( {\omega/\omega_{ni}} \right)} \right\rbrack^{2}}}{\left( {\omega/\omega_{ni}} \right)^{2}}\quad\lbrack{cm}\rbrack}}} & {3\text{-}2}\end{matrix}$

4. The direction of vector F₁, F₂, namely the angular position θ1, θ2 tothe X-coordinate (basis) can be obtained by that the phase angles: αiare revised to the measured angular positions of the vibration vectorsX₁, X₂ (displacement) for the F₁, F₂ (force) [N].

αi are expressed by the following equation $\begin{matrix}{\alpha_{i} = {\tan^{- 1}{\frac{2\left( {\omega/\omega_{ni}} \right)\left( {c_{1}/c_{oi}} \right)}{1 - \left( {\omega/\omega_{ni}} \right)^{2}}\quad\lbrack{rad}\rbrack}}} & {4\text{-}1}\end{matrix}$

5. Determining the dynamic unbalance of the section, b, of rotor R basedon the quantities of the unbalance centrifugal force vector F₁, F₂(magnitude, direction)at the bearings B1,B2 (journal portions) obtainedby Eq. 1-4 or Eq. 2-1, can be calculated as following descriptions basedon the view point of the feature of the invention that it is similar tomove or change the arbitrary dynamic balance planes.

Where, the both sides of the section, b, of rotor R are two correctionplanes I, II (this means that the length between the correction planes Iand II is ,b,) And, let the lengths between the both correction planesand the both bearings B1,B2 are a₁ and a₂ [cm], namelyRatio of distances between bearings and correction-planes=a₁:b:a₂[cm]  5-1

6. In case of a₁=a₂ and F₁≈F₂:

(the surface of FIG. 2 means that X-coordinate and Y-coordinate arecrossing on the face, and Z-coordinate is perpendicular to the surfaceof FIG. 2)

The magnitudes of vectors F₁, F₂ [N] obtained by Eq. 1-4 or 2-1, 3-2,are converted to the arbitrary lengths and drew on FIG. 2.

And the length of straight line which is connected between the tips ofarrows of vectors F₁, F₂ on FIG. 2, is drew as b′, then the line b′ areextended to the both outside as the a₁′ of F₁ side and a₂′ of F₂ side,and the ratio of lengths of 3-line shall be set as equal to the ratio ofdistances between the bearings-correction-planes of Eq. 5-1.

Namely,a₁′:b′:a₂′=a₁:b:a₂ [cm]  6-1

7. Next on the FIG. 2, the lines F_(I), F_(II) are drew by connectingwith between the each tips of lines a₁′, a₂′ and the center ofrotational axis Z, are obtained as the dynamic unbalance vectors (N,rad) of correction planes I, II in rotor that corresponding to F₁, F₂(dynamic unbalance centrifugal forces) vectors (N,rad)

8. The magnitudes [N] of dynamic unbalance centrifugal force vectorsF_(I), F_(II) can be obtained by the following equation based on theratio of arbitrary unit length on the FIG. 2 $\begin{matrix}{F_{I,{II}} = {F_{1,2}{\frac{{length}\quad{of}\quad{vector}\quad F_{I,{II}}}{{length}\quad{of}\quad{vector}\quad F_{1,2}}\quad\lbrack N\rbrack}}} & {8\text{-}1}\end{matrix}$

9. The amount of specific unbalance, e_(I,II)[cm] corresponding to thedynamic unbalances F_(I), F_(II)[N] are can be obtained by that theresult of Eq. 8-1 is substituted in the following equation$\begin{matrix}{{{e_{I} = \frac{F_{I}}{M_{I}\omega^{2}}},{e_{II} = {\frac{F_{II}}{M_{II}\omega^{2}}\quad\lbrack{cm}\rbrack}}}{{Where},}} & {{9\text{-}1},{9\text{-}2}} \\{M_{I},{{M_{II}\text{:}\quad M_{I}} = {\frac{h_{II}}{b}M}},\quad{M_{II} = {\frac{h_{I}}{b}{M\quad\left\lbrack {{Ns}^{2}/{cm}} \right\rbrack}}}} & {{9\text{-}3},{9\text{-}4}}\end{matrix}$

10. Dynamic unbalance in the case of change for the distance, b, betweentwo correction planes I, II in the rotor on FIG. 1, (which will bechanged to half distance ½b), on FIG. 2, the length: b′ is not changed,the line at F₁ side will be extended as a₁″ and also the line at F₂ sidewill be extended as a₂″. As result, the ratio of the three changed linesshall be satisfied within the ratio of following equation.

Namely,a ₁ ″:b′:a ₂″=(a ₁+(¼)b):(½)b:(a ₂+(¼)b)   10-1

11. Then, on FIG. 2, the each tips of the lines a₁″, a₂″ and the centerof rotational axis, Z are connected, and the lines F_(I)′, F_(II)′ canbe produced.

The lines F_(I)′, F_(II)′ are obtained as the dynamic unbalance vectors(ratio of magnitude, angle position) corresponding to F₁, F₂ (unbalancecentrifugal forces) [N,rad] at the correction planes I, II, in case ofthe distance between 2-correction planes in rotor has been changed to ½of ,b,

12. The amounts of specific unbalance of dynamic unbalance, e_(I)′,e_(II)′ [cm] in case of the distance: ½b between the 2-correction planeswhich can be obtained by the simulating calculation in accordance withEqs. 8-1, 9-1, and 9-2.

13. In case of a₁≠a₂:

The calculation to determine the dynamic unbalance can be performed thatthe dynamic unbalance vectors: F_(I), F_(II)[N] obtained by above Par.6˜9 (in case of a₁=a₂) are distributed to the designed correction planesin accordance with the conventional method of statics, and the dynamicunbalance vectors of the each designed plane can be determined by thegeometric re-addition of these distributed vectors.

And in case of the directly calculating to determine the dynamicunbalance from the unbalance centrifugal forces vectors F₁, F₂ [N],based on the view-point “the planes of both journals (bearings) are alsoequal to dynamic balance plane in rotor”, the dynamic unbalance vectorsF_(I), F_(II) [N] of rotor can be determined in following Pars. by thevector-geometric-calculation using principles of statics as showing inFIG. 3 ( three-dimensional drawing) and FIG. 4.

14. At first, as showing FIG. 3 (three-dimensional drawing), theoriginal magnitude F₁, F₂ [N] are distributed to the designed dynamicbalance plane (correction planes) I, II by the following equation

And the magnitudes of component vectors can be obtained.

-   -   component vectors on F₁ plane: $\begin{matrix}        {{F_{1I} = {F_{1}\frac{a_{1} + b}{b}}},{F_{2I} = {F_{2{II}}{\frac{a_{2}}{b + a_{2}}\quad\lbrack N\rbrack}}}} & \text{14-1,14-2}        \end{matrix}$    -   component vectors on F_(II) plane: $\begin{matrix}        {{F_{1{II}} = {F_{1I}\frac{a_{1}}{a_{1} + b}}},{F_{2{II}} = {F_{2}{\frac{b + a_{2}}{b}\quad\lbrack N\rbrack}}}} & \text{14-3,14-4}        \end{matrix}$

15. Next, the resultant vector F₁ from the component vectors of Eqs.14-1,2 and the resultant vector F_(II) from the component vectors ofEqs. 14-3,4 are showed on FIG. 4 as the dynamic unbalance vectors of thedynamic balance planes I, II, namely, the dynamic unbalance vectorsF_(I), F_(II) of the dynamic balance planes I, II can be obtained by thevector-calculation based on “parallelogram law” showed in FIG. 4 or thefollowing equations according to the formulas of “triangle of force”.Namely,F _(I)=√{square root over (F _(1I) ² +F _(2I) ²−2F _(1I) F _(2I) COS(180−τ))} [N]  15-1F _(II)=√{square root over (F _(1II) ² +F _(2II) ²−2F _(1II) F _(2II)COS (180−τ))} [N]  15-2provided that,τ=(θ₁+180)−θ₂ [deg]  15-3

16. The amounts of specific unbalance e_(I), e_(II) [cm] correspondingto the dynamic unbalance centrifugal force F_(I), F_(II) [N] determinedin the above Par. 15, which can be obtained by simulating in accordancewith Eqs. 8-1,9-1,and 9-2. Showing in the above descriptions of “Bestmode of carrying out the invention”, the calculation procedures with theequations and geometric calculations with vector in FIGS. 2 and 4, whichare the features of this invention.

And the numeric calculation used with the trigonometrical function andthe sine and cosine law of trigonometry that simulated with thegeometric calculations with vector in FIGS. 2 and 4, which are alsovalid in this invention.

And these calculation mode can be applied the vectors measured for thevelocity and acceleration and for the computer of field balancing anddynamic balancing machines, which are also valid in this invention.

Possibility on the Industrial Utilization:

This invention can directly calculate to determine the dynamic unbalanceof rotating machine from the original vibration vectors detected at thebearings. Therefor the test runs which are subject to detect the changeseffected with the trial masses in field balancing, they are unnecessary,so the dynamc balancing works can be simplified and reduced in measuringtime.

The merits due to reduction increase according to the more larger sizerotor.

And this invention makes easily to correct the unbalance, to testbalancing and to monitor for the unbalance vibration in the areas of thechemical plants, nuclear power plants and such plants using dangerousobjects, and to contribute for the safety. On the dynamic balancing testat the shop of rotating machinery, the application of this inventionmakes to reduce the measuring times and to simplify the methods andprocedures for measuring, and to be valid to reduce for burdens of thepowers, labors, and equipments.

1. The calculating method to determine the dynamic unbalance in rotorsby the vector calculation which is using with the ratios between thedistance lengths of bearings of machines and apparatus and the ratios ofthe distance length of arbitrarily chosen two correction planes withoutthe test runs attached with trial masses, based on the data consist ofmagnitudes and directions for the vibration vector quantities ofdisplacement, velocity, or acceleration which originated in dynamicunbalance in rotor, and which are measured at the two bearings of therotating machines or dynamic balancing machines
 2. The computers,measuring instruments and testing facilities, which are directly orindirectly applied or equipped the calculating method to determine thedynamic unbalance in rotor with the vector calculation mentioned in theclaim 1.